Vivek has provided a great answer, but here is something else. Most of the time, to really extract the kind of detail and color fidelity you need for underwater photography, you want to use the underwater filtration. This usually means using a lot of magenta filtration to counter the strong blue tint. How much filtration you use depends on the depth, the deeper you go the more you tend to need. You can try to use just RAW and post-process WB correction, however filtration actually blocks excess blue light, usually providing better results (even if you still correct in post.)
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jrista♦Sep 25 '12 at 18:19

3 Answers
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Can be used, but need a strong source of light - without artificial light, it is useless (see the wiki answer of glass squid higher). Also all the floating particles between you and subjects will be highly visible, so still need to be as close as possible.

The only source of light under water is the light that is refracted from the outside. Also there is a fact that the light that is refracted into the water surface will be partially polarized.Partially polarized meaning that the refracted light has all planes of electromagnetic vibration other than the plane of vibration of the reflected light at the surface of water. As explained by below diagram:

source:wikipedia.

So from this we can understand that any light under the water will have all planes of vibration except the plane parallel to the surface of the water.

Hence using a polarizer that acts in the plane parallel to water surface will allow no light (theoretically) but a very low amount of light (practically). Using the polarizer in any other angle will result in no notable change.
From the explanation, circular polarization on underwater photography is possible. But in my opinion its more of useless. There is no real use of it in underwater photography. So it is not used for underwater.

The assumption that the only light comes from above water seems flawed. I found multiple models of underwater strobes quite easily, and artificial lighting seems to be the only way to see anything deeper below. Using a polarizing filter to kill the ambient light in order to give preference to photographer's own light could be quite useful.
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ImreDec 4 '12 at 20:24